02/28/2010, 12:22 AM
im thinking about using asymt of exp(x) to solve tetration.
perhaps not the best one , but sinh(x) comes to my mind.
if we take the half-iterate of sinh(x) by using taylor series , we get a good approximation of the half-iterate of exp(x) for x large.
if this good approximation is analytic in say [e^e,e^e^e^e] we could use that interval and take logs or exp of it to compute the half-iterate for [-oo,+oo] up to a relatively high precision.
in fact , i assume , we can choose our precision if the good approximation is entire , by using bigger intervals and taking logs of it ( as somewhat done above ).
we might even consider the mittag-leffler expansion to avoid problems with non-analytic issues ( thus taking mittag-leffler expansion of the formal taylor series )
thats basicly the idea , but as said maybe sinh(x) isnt the best , on the other hand its probably the easiest.
furthermore some want - and me too actually - that half-iterates have non-negative (2+n)th derivatives and strictly positive zeroth , first and second derivatives.
that last restriction troubles me , especially when i try to get closer to exp(x) than sinh(x) by using function that have non-negative derivatives ... getting conditions on a particular n'th derivate isnt hard but the whole problem is more troublesome ( maybe someone knows a solution to this ? )
i didnt mention that all the above ofcourse is about functions going from R -> R , its obvious , but i just mention it to avoid potential confusion.
also every approximation of exp(x) should equal x at x = 0 only and be larger than id(x).
hope its clear.
maybe you considered this once too ?
greetings my fellow euh ... tetrationalists
regards
tommy1729
ps : showing that all derivatives are positive can sometimes be easy , but in general its hard , see for example " li's criterion " for the RH.
maybe its easy here and i missed a trivial thing ... ( im getting old ? :p )
perhaps not the best one , but sinh(x) comes to my mind.
if we take the half-iterate of sinh(x) by using taylor series , we get a good approximation of the half-iterate of exp(x) for x large.
if this good approximation is analytic in say [e^e,e^e^e^e] we could use that interval and take logs or exp of it to compute the half-iterate for [-oo,+oo] up to a relatively high precision.
in fact , i assume , we can choose our precision if the good approximation is entire , by using bigger intervals and taking logs of it ( as somewhat done above ).
we might even consider the mittag-leffler expansion to avoid problems with non-analytic issues ( thus taking mittag-leffler expansion of the formal taylor series )
thats basicly the idea , but as said maybe sinh(x) isnt the best , on the other hand its probably the easiest.
furthermore some want - and me too actually - that half-iterates have non-negative (2+n)th derivatives and strictly positive zeroth , first and second derivatives.
that last restriction troubles me , especially when i try to get closer to exp(x) than sinh(x) by using function that have non-negative derivatives ... getting conditions on a particular n'th derivate isnt hard but the whole problem is more troublesome ( maybe someone knows a solution to this ? )
i didnt mention that all the above ofcourse is about functions going from R -> R , its obvious , but i just mention it to avoid potential confusion.
also every approximation of exp(x) should equal x at x = 0 only and be larger than id(x).
hope its clear.
maybe you considered this once too ?
greetings my fellow euh ... tetrationalists
regards
tommy1729
ps : showing that all derivatives are positive can sometimes be easy , but in general its hard , see for example " li's criterion " for the RH.
maybe its easy here and i missed a trivial thing ... ( im getting old ? :p )
